Different implicit copulas can be constructed by using different conditionally Gaussian priors for \(\text{$\beta $}\)  at (REF ).
[1]} explore three different choices, including the horseshoe prior of [2]} which
is outlined here. This prior
provides regularization of \(\text{$\beta $}\)  in the auxiliary regression.
The prior is given by
\(\text{$\beta $}|\text{$\lambda $},\tau &\sim &N\left(\mathbf {0},\mbox{diag}(\text{$\lambda $})^2\right)\,,\;\text{$\lambda $}=(\lambda _1,\ldots ,\lambda _p)^\top \,,\\\lambda _j|\tau &\sim &\mbox{Half-Cauchy}(0,\tau )\,,\mbox{ and }\tau \sim \mbox{Half-Cauchy}(0,1)\,,\) 
